Nonlinear Optics
During the last years we have focused on the mathematical backgrounds of dispersion management and related problems. The model equation describing pulse propagation in dispersion compensated transmission systems is the nonlinear Schrödinger equation with rapidly varying periodic coefficients, one is interested in periodic solutions of the NLS called breathing (or DM) solitons. We were able to establish the existence of ground states for some suitable averaged version of the master equation by proving a general theorem on the bifurcation of solutions from the essential spectrum without sign condition on the nonlinearity. As a further practical application we have studied the NLS with an additional quadratic potential of trapping type, which is the appropriate equation for pulse propagation in transmission systems with in-line phase modulators. Due to the potential there exist bound states with Gaussian tails. Such pulses reduce soliton interaction and are very useful for more dense information packing. Another interesting approach is to look for pulses with self-similar properties. In this context we proved the existence of periodic solutions of a singular Lagrangian system that describes the evolution of optical pulse width and chirp. Furthermore we have derived a new Schrödinger-type equation with additional quadratic potential which describes pulse evolution after lens transformation and can be averaged. In the case of strong dispersion management the resulting equation has an additional quadratic potential of trapping type. Due to potential we can characterize both, local and global bifurcation behaviour.
References:
T. KÜPPER, C. JONES, K. SCHAFFNER: Bifurcation of asymmetric solutions in nonlinear optical media, submitted to ZAMP
M. KUNZE, T. KÜPPER, V.K. MEZENTSEV, E.G. SHAPIRO & S. TURITSYN: Nonlinear solitary waves with Gaussian tails, Physica D 128, 273-295 (1999)
M. KUNZE: Periodic solutions of a singular Lagrangian system related to dispersion-managed fiber communication devices, submitted to Nonlinear Dynamics and Systems Theory
M. KUNZE: Bifurcation from the essential spectrum without sign condition on the nonlinearity for a problem arising in optical fiber communication systems, submitted to Proc. Roy. Soc. Edinburgh, Ser. A
M. KURTH: Nonlinear Schrödinger equations in fiber optics, Diploma thesis, Cologne (1999) (German)
M. KURTH: Optical Solitons as Ground States of NLS in the Regime of Strong Dispersion Management, accepted by Physica D
M. KURTH: Bifurcation analysis of an NLS with quadratic potential related to dispersion managed solitons, in preparation
M. KURTH: Verzweigung von DM-Solitonen in optischen Übertragungssystemen, PhD thesis, Cologne (2003) Download
People: Tassilo Küpper, Michael Kurth, Markus Kunze (former member)
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